Convergence Properties of Monotone and Nonmonotone Proximal Gradient Methods Revisited

TL;DR

This paper revisits convergence properties of proximal gradient methods for composite optimization, providing new results that do not require the often restrictive Lipschitz continuity assumption on the smooth part.

Contribution

It offers convergence analysis for both monotone and nonmonotone proximal gradient methods without assuming Lipschitz continuity of the gradient.

Findings

Convergence results applicable without Lipschitz assumptions

Analysis omits descent lemmas, allowing broader applicability

Applicable to practical scenarios with non-Lipschitz smooth functions

Abstract

Composite optimization problems, where the sum of a smooth and a merely lower semicontinuous function has to be minimized, are often tackled numerically by means of proximal gradient methods as soon as the lower semicontinuous part of the objective function is of simple enough structure. The available convergence theory associated with these methods (mostly) requires the derivative of the smooth part of the objective function to be (globally) Lipschitz continuous, and this might be a restrictive assumption in some practically relevant scenarios. In this paper, we readdress this classical topic and provide convergence results for the classical (monotone) proximal gradient method and one of its nonmonotone extensions which are applicable in the absence of (strong) Lipschitz assumptions. This is possible since, for the price of forgoing convergence rates, we omit the use of descent-type lemmas in our analysis.